Nuprl Lemma : qabs-difference-zero
∀[r,s:ℚ]. uiff(|r - s| ≤ 0;r = s ∈ ℚ)
Proof
Definitions occuring in Statement :
qabs: |r|,
qle: r ≤ s,
qsub: r - s,
rationals: ℚ,
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
natural_number: $n,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
prop: ℙ,
subtype_rel: A ⊆r B,
implies: P ⇒ Q,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
true: True,
qsub: r - s,
squash: ↓T,
guard: {T}
Lemmas referenced :
equal-wf-T-base,
rationals_wf,
qsub_wf,
equal_wf,
iff_weakening_uiff,
qle_wf,
qabs_wf,
qabs-qle-zero,
qle_witness,
uiff_wf,
int-subtype-rationals,
qadd_wf,
qmul_wf,
squash_wf,
true_wf,
qadd_ac_1_q,
qadd_comm_q,
qinverse_q,
mon_ident_q,
iff_weakening_equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
independent_pairFormation,
hypothesis,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
baseClosed,
addLevel,
productElimination,
independent_isectElimination,
natural_numberEquality,
applyEquality,
because_Cache,
sqequalRule,
independent_functionElimination,
cumulativity,
independent_pairEquality,
isect_memberEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
applyLambdaEquality,
minusEquality,
lambdaEquality,
imageElimination,
universeEquality,
imageMemberEquality,
hyp_replacement
Latex:
\mforall{}[r,s:\mBbbQ{}]. uiff(|r - s| \mleq{} 0;r = s)
Date html generated:
2018_05_21-PM-11_53_00
Last ObjectModification:
2017_07_26-PM-06_45_20
Theory : rationals
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